Simplify (-1/2x)(4x^2)^3

$(- \frac{1}{2} x) {(4 x^{2})}^{3}$

Step 1

Combine $x$ and $\frac{1}{2}$ .

$- \frac{x}{2} {(4 x^{2})}^{3}$

Step 2

Simplify the expression.

Apply the product rule to $4 x^{2}$ .

$- \frac{x}{2} (4^{3} {(x^{2})}^{3})$

Raise $4$ to the power of $3$ .

$- \frac{x}{2} (64 {(x^{2})}^{3})$

Step 3

Cancel the common factor of $2$ .

Move the leading negative in $- \frac{x}{2}$ into the numerator.

$\frac{- x}{2} (64 {(x^{2})}^{3})$

Factor $2$ out of $64 {(x^{2})}^{3}$ .

$\frac{- x}{2} (2 (32 {(x^{2})}^{3}))$

Cancel the common factor.

$\frac{- x}{2} (2 (32 {(x^{2})}^{3}))$

Rewrite the expression.

$- x (32 {(x^{2})}^{3})$

Step 4

Simplify the expression.

Multiply $32$ by $- 1$ .

$- 32 x {(x^{2})}^{3}$

Multiply the exponents in ${(x^{2})}^{3}$ .

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$- 32 x \cdot x^{2 \cdot 3}$

Multiply $2$ by $3$ .

$- 32 x \cdot x^{6}$

Step 5

Multiply $x$ by $x^{6}$ by adding the exponents.

Move $x^{6}$ .

$- 32 (x^{6} x)$

Multiply $x^{6}$ by $x$ .

Raise $x$ to the power of $1$ .

$- 32 (x^{6} x^{1})$

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$- 32 x^{6 + 1}$

Add $6$ and $1$ .

$- 32 x^{7}$